Karlsruhe Institute of TechnologyInstitute of Engineering Mechanics

University of PaderbornChair of Engineering Mechanics

GRK 2078 International Research Training Group

29th International Workshop

Research in Mechanics of Composites

Bad Herrenalb, GermanyDecember 6-8, 2016

Objective of the Workshop

Modern fiber reinforced polymers (FRP) show a macroscopic material behaviordepending sensitively on the fiber orientation distribution and arrangement as wellas on the generally nonlinear material behavior of the constituents. Additionally, theoverall composite behavior is influenced by fluid-structure interaction, by curingduring the production process as well as by the interface properties. Understandingthe correlation of both the microstructure and the micromechanical behavior onthe one side, and the macroscopic composite behavior on the other side, is offundamental interest for the design of materials, the optimization of productionprocesses as well as the dimensioning and optimization of construction parts.In this workshop, new approaches for the material modeling of short and longfiber reinforced composites, corresponding numerical solution strategies, andexperimental techniques are discussed. Special emphasis is given to the modeling ofprocess chains. The International Research Training Group "Integrated Engineeringof continuous-discontinuous long fiber reinforced polymer structures", funded by theGerman Research Foundation (DFG), provides a structured educational program forseveral PhD students focusing on research in the areas of materials science, productengineering, mechanics, production science and light-weight technologies.

The Organizers

Prof. Dr.-Ing. habil. Thomas BöhlkeProf. Dr.-Ing. habil. Rolf MahnkenGRK 2078 International Research Training Group

Graphic on front page:

Reference-graphic by Pascal Pinter, A novel Method for Determination of Fiber Length Distributions from µCT-data,

P. Pinter, B. Bertram, K. Weidenmann (KIT), Karlsruhe, Germany

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Effective and efficient modeling of viscous short-fiberreinforced polymer flow

Róbert Bertóti and Thomas Böhlke

Chair for Continuum Mechanics,Institute of Engineering Mechanics, Karlsruhe Institute of Technology (KIT), Karlsruhe, Germany

Abstract. For the effective modeling of viscous short-fiber reinforced polymer flow,so called Fiber Orientation Tensors (FOTs) are used [1]. The FOTs describe themicrostructure of a fluid-fiber mixture on the macro scale in an effective manner. Inthe current work, Fiber Orientation Vectors (FOVs) are also used which describe themicrostructure of the fluid-fiber mixture on the micro scale in a discrete manner. Theevolution equations of the FOVs and the FOTs are all based on Jeffery’s equation [2].The effectiveness and efficiency of the FOTs vs. the FOVs is shown in this work on theexample of a numerical implementation for simple flow cases.

Not only the fluid-fiber interaction, but also the fiber-fluid interaction is described inan effective way. A two step homogenization method, based on a Hashin-Shtrikmanbound [3], is used to calculate the flow induced anisotropic viscosity of the fluid-fibermixture. This method is compared numerically to the Dinh-Armstrong constitutivemodel [4] by means of simple flow cases.

References

[1] S. G. Advani, C. L. Tucker III: The Use of Tensors to Describe and Predict FiberOrientation in Short Fiber Composites, Journal of Rheology, 31(8), 751–784 (1987).

[2] G. B. Jeffery: The Motion of Ellipsoidal Particles Immersed in a Viscous Fluid,Proceedings of the Royal Society of London, Series A 102(715), 161–179 (1922).

[3] J. R. Willis: Bounds and self-consistent estimates for the overall properties ofanisotropic composites, J. Mech. Phys. Solids, 25, 185–202 (1977).

[4] A. Latz, U. Strautins, D. Niedziela: Comparative numerical study of twoconcentrated fiber suspension models, Journal of Non-Newtonian fluid mechanics,165(13-14), 764–781 (2010).

(n)- and (n + 1)-layered composite sphere models forthermo-chemo-mechanical effective properties

Christian Dammann, Rolf Mahnken, and Peter Lenz

Chair of Engineering Mechanics,Paderborn University, Paderborn, Germany

Abstract. Our work presents extensions of multi layered composite sphere models knownfrom the literature [1, 2, 3] to temperature-dependent elastic effects accompaniedby curing. Effective properties in dependence on the degree of cure are obtainedby homogenization for a representative unit cell (micro-RVE) on the heterogeneousmicroscale. To this end, analytical solutions for (n)- and (n + 1)-layered compositesphere models, [3, 4], are derived, in addition to Voigt and Reuss bounds resultingfrom the assumption of a homogeneous mixture. For simplification, we restrict thematerial behavior of the micro-RVE to a thermo-chemo-mechanical coupling withlinear elasticity. In a numerical study we compare different effective material propertiesincluding bounds.

References

[1] Z. Hashin: The elastic moduli of heterogeneous materials, J. Appl. Mech., 29, 143–150(1962).

[2] R. M. Christensen: Mechanics of composite materials. John Wiley and Sons, New York,1979

[3] E. Herve, A. Zaoui: n-layered inclusion based micromechanical modeling, Int. J.Eng. Sc., 31, 1–10 (1993).

[4] A. Gusev: Effective coefficient of thermal expansion of n-layered composite spheremodel: Exact solution and its finite element validation, Int. J. Eng. Sc., 84, 54–61(2014).

Transformation strains and variant interaction for bainiticvariant evolution.

Ulrich Ehlenbröker1, Manuel Petersmann2, Thomas Antretter2, and Rolf Mahnken1

1 Chair of Engineering Mechanics,Paderborn University , Paderborn, Germany

2 Institute of Mechanics,Montanuniversität Leoben, Leoben, Austria

Abstract. The evolution of the baintic phase in steel is of particular relevanceespecially in industrial production processes encompassing hot-forming and quench-ing of larger components. For that reason a multi-scale model for bainitic phasetransformation in multi-variant polycrystalline low alloy steels has been developed[1].

In order to yield relasitic simulation results, the microscopic conversion proceduresfor the austenite-to-bainite transformation have to be described in an appropriate way.To that end, the transformation strains for the crystallographic variants in the modelhave been adjusted. For the calculation of transformation strains, different theoriescan be applied, e.g. a hierarchical block-structure as described in [2] for the martensticphase transformation in a 9Ni-steel, or a theory solely based on the lattice parametersof the two phases (austenite and bainite) and the assumption that the transformationleaves a close packed plane and a close packed direction in that plane unrotated, inother words the measured orientation relationship for the transformation (see [3]).Further, a mechanism for the simulation of variant interaction between the differentcrystallographic variants, based on the theory of transformation hardening, has beenintroduced into the model (see [4]).

References

[1] R. Mahnken, A. Schneidt, T. Antretter, U. Ehlenbröker, M. Wolff: Multi-scalemodeling of bainitic phase transformation in multi-variant polycrystalline low alloysteels, Int. J. Solids Struct., 54, 156–171 (2015).

[2] L. Qi, A.G. Khachaturyan, J.W. Morris Jr.: The microstructure of dislocatedmartensitic steel: Theory, Acta Mater., 76, 23–39 (2014).

[3] K. Koumatos, A. Muehlemann: A Theoretical Investigation of Orientation Rela-tionships and Transformation Strains in Steels, [cond-mat.mtrl-sci], arXiv:1604.05270,(2016).

[4] U. Ehlenbröker, R. Mahnken, M. Petersmann, T. Antretter: Modeling of variant-interaction during bainitic phase transformation, IOP Conf. Series: Mater Sci. Eng.,119, (2016).

Consistent macroscopic tangent computation forFFT-based homogenization

Felix Selim Göküzüm and Marc-André Keip

Institute of Applied Mechanics (Civil Engineering), Chair IUniversity of Stuttgart, Stuttgart, Germany

Abstract. Lately, the FFT-based method suggested by MOULINEC & SUQUET [1] hasincreased in popularity due to its superior computational speed. The present workgives a computational framework for evaluating the consistent tangent for FFT-basedhomogenization. It is inspired by the works on multiscale FEM, see e.g. MIEHE ET AL.[2]. The proposed consistent tangent for FFT-based approaches offers an alternative tonumerical tangents calculation, which are carried out by perturbing the single straincomponents. Especially in the context of multi-scale methods, e.g. FE2 or FE-FFT [3,4],the computation of a consistent macroscopic tangent is crucial. Computing the tangentnumerically can be costly with respect to computational time and might even surpassthe computational time needed for solving the initial mechanical equilibrium. We willshow that the use of an algorithmic tangent offers a more efficient and faster calculationof the macroscopic tangent, however, it comes with the drawback of an increasedmemory storage demand. The viability of the method holds for linear as well as fornonlinear material behaviour, including viscoelasticity and large strains as discussedin KABEL ET AL. [5].

References

[1] H. Moulinec, P. Suquet: A fast numerical method for computing the linear andnonlinear mechanical properties of composites, Académie des Sciences, 2, 1417–1423(1994).

[2] C. Miehe, J. Schotte, J. Schröder: Computational micro-macro transitions andoverall moduli in the analysis of polycrystals at large strains, ComputationalMaterials Science, 16, 372–382 (1999).

[3] J. Spahn, H. Andrä, M. Kabel, R. Müller: A multiscale approach for modelingprogressive damage of composite materials using fast Fourier transforms, ComputerMethods in Applied Mechanics and Engineering, 268, 871–883 (2014).

[4] J. Kochmann, S. Wulfinghoff, S. Reese, J. R. Mianroodi, B. Svendsen: Two-scaleFE-FFT- and phase-field-based computational modeling of bulk microstructuralevolution and macroscopic material behavior, Computer Methods in AppliedMechanics and Engineering, 305, 89–110 (2016).

[5] M. Kabel, T. Böhlke, M. Schneider: Efficient fixed point and Newton-Krylov solversfor FFT-based homogenization of elasticity at large strains, Computational Mechanics,54, 1497–1514 (2014).

Considering the fiber orientation for the computation ofthe hyperelastic Tucker average for short fiber reinforcedcomposites

Niels Goldberg1, Felix Ospald2, Matti Schneider3, and Jörn Ihlemann1

1 Professorship of Solid Mechanics, Faculty of Mechanical Engineering,TU Chemnitz, Chemnitz, Germany

2 Professorship of Partial Differential Equations, Faculty of Mathematics,TU Chemnitz, Chemnitz, Germany

3 Fraunhofer ITWM, Kaiserslautern, Germany

Abstract. In this contribution we apply a fiber orientation-adapted integration schemeto the computation of the Tucker average of non-linear material laws for shortfiber reinforced composites based on angular central Gaussian fiber orientationdistributions. We establish a reference scenario for fitting the Tucker average of atransversely isotropic hyperelastic energy to microstructural simulations, obtained byFFT-based computational homogenization. We carefully discuss ideas for acceleratingthe identification process, leading to a tremendous speed-up compared to a naiveapproach. The resulting hyperelastic material map turns out to be surprisinglyaccurate, simple to integrate in commercial finite element software and fast in itsexecution. We demonstrate the capabilities of the extracted model by a finite elementanalysis of a fiber reinforced chain link.

Stress-strain characterization and damage modeling ofglass fiber reinforced polymer composites with vinylestermatrix

Jonas Hund1, Christian Leppin2, Thomas Böhlke3, and Jan Rothe2

1 Institute of Mechanics,Karlsruhe Institute of Technology (KIT), Karlsruhe, Germany

2 Suisse Technology Partners AG,Neuhausen am Rheinfall, Switzerland

3 Institute of Engineering Mechanics,Karlsruhe Institute of Technology (KIT), Karlsruhe, Germany

Abstract. Beyond a certain loading threshold, glass fiber reinforced polymer compos-ites with vinylester matrix exhibit considerably nonlinear deformation behavior dueto damage of the matrix material. To model such nonlinear deformation properties,an approach of a fully anisotropic damage model suggested by Govindjee et al.[1]is considered. In addition, the inter-fiber fracture criterion introduced by Puck[2] isused for the damage function which defines the damage initiation and the materialstrength. For identifying the material properties as well as validating the modelthree GFRP laminates with different layups are tested in tension under differentloading orientations, assuming laminate theory to be reasonably well fulfilled. Furtherexperiments are compared with corresponding simulation results to demonstrate theperformance of the model.

References

[1] Govindjee, S., Kay, G.J., Simo, J.C.: Anisotropic modelling and numerical simulationof brittle damage in concrete, International Journal for Numerical Methods inEngineering, 38, 3611–3633 (1995).

[2] Puck, A. Festigkeitsanalyse von Faser-Matrix-Laminaten: Modelle für die Praxis. HanserFachbuchverlag, Munich, 1996

Efficient hourglass control in FFT-based computationalhomogenization

Matthias Kabel and Matti Schneider

Department of Flow and Material Simulation, Fraunhofer ITWM, Kaiserslautern

Abstract. The FFT-based homogenization method of Moulinec and Suquet [1] is a fastand memory-efficient solver posed on regular voxels grids and is well adapted forelastic simulations on complex microstructures of fiber reinforced plastics. Originally,the method used Fourier polynomials to discretize the continuous version of theLippmann-Schwinger equation, which lead to convergence problems at high materialcontrasts (e.g. pores and defects). Recently, many groups are working on usingalternative discretizations in combination with the FFT-based method: Voxelwiseconstant strains [2], finite differences on a staggered grid [3], trilinear hexahedralelements with reduced integration [4] and trilinear elements with full integration [5].

All of these discretization have major drawbacks: No explicit formula for the Green’soperator, material function has to be evaluated on a double fine grid, hourglassingor high computational effort. In our talk we will propose a computational efficientway to insert an artificial stiffness to the hourglass deformation modes [6] utilizingthe reference material and assess the solution quality as well as the computationalefficiency of this new discretization.

References

[1] H. Moulinec and P. Suquet. A numerical method for computing the overall responseof nonlinear composites with complex microstructure. Computer Methods in AppliedMechanics and Engineering, 157(12),69 – 94 (1998)

[2] S. Brisard and L. Dormieux. FFT-based methods for the mechanics of composites: Ageneral variational framework. Computational Materials Science 49(3), 663–671 (2010)

[3] M. Schneider, F. Ospald and M. Kabel. Computational homogenization of elasticityon a staggered grid. International Journal for Numerical Methods in Engineering 105(9),693 – 720 (2016)

[4] F. Willot. Fourier-based schemes for computing the mechanical response ofcomposites with accurate local fields. Comptes Rendus Mécanique 343(3), 232–245(2015)

[5] M. Schneider, D. Merkert, M. Kabel. FFT-based homogenization for microstructuresdiscretized by linear hexahedral elements. International Journal for NumericalMethods in Engineering (2016)

[6] T. Belytschko, J. S.-J. Ong, W. K. Liu, J. M. Kennedy. Hourglass control in linearand nonlinear problems, Computer Methods in Applied Mechanics and Engineering,43(3),251–276 (1984)

Questions regarding the development of mappingalgorithms for short fiber reinforced composite modeling

Christian Liebold1, Dr. André Haufe1, and Prof. Dr.-Ing. Peter Middendorf2

1 DYNAmore GmbH,Industriestraße 2, Stuttgart, Germany

2 Institute of Aircraft Design,University of Stuttgart, Germany

Abstract. This work will emphasis the importance of a proper data transformationfrom the process simulation to the finite element mesh for structural analysis.As an example, two modeling approaches for short fiber reinforced compositeswithin a commercial finite element code will be compared. Thereby, differenthomogenization, averaging and interpolation techniques will be introduced anddiscussed. In [1], interpolation and averaging techniques for scalar values such asstrains and thicknesses based on the finite element integration methods have beendeveloped, not covering the topic of a proper consideration of tensorial data. This iscrucial since standard averaging methods will lead to the loss of information aboutthe tensors’ shape being described by its eigenvectors and eigenvalues and thereby,the loss of information about the degree of anisotropy. A method to overcome thisproblem is introduced in [2] and will be discussed within this work. Averaging andinterpolation techniques for scalar values such as they are described in [3] will beintroduced as well, trying to identify the most reliable averaging method. The gainedknowledge about proper data transformation and homogenization shall be applicablefor further processing techniques such as continuous fiber reinforced composites ormetal forming.

The influence of scattering data from process simulation on the results of structuralanalysis will be discussed as well.

References

[1] K. Wolf, U. Scholl, P. Post, J.-V. Peetz, M. Ottavio, T. Wallmersperger,M. Waedt, B. Kröplin: Verbesserung der Prognosefähigkeit der Crashsimulationaus höherfesten Mehrphasenstählen durch Berücksichtigung von Ergebnissenvorangestellter Umformsimulationen. In: Forschungsreihe Automobiltechnik e.V. - FATSchriftenreihe 198, Frankfurt, GER, 2005

[2] J. K. Gahm: Microstructural feature-based Processing and Analysis of Diffusion TensorMRI. University of California, USA, 2014

[3] D. Shepard: A two-dimensional interpolation function for irregularly-spaced dataIn: Proceeding ACM National Conference, 1968

Investigating the fibre length distribution from CTimages using fibre endpoints

Jan Niedermeyer and Claudia Redenbach

TU Kaiserslautern, Kaiserslautern, Germany

Abstract. Estimating the fibre length distribution in fibre reeinforces polymers fromCT data is still a challenge. Image quality usually does not allow a full fibresegmentation.

Using Gaussian curvature we can segment the fibre endpoints from a CT image. Byinterpreting these endpoints as a point process we can then use point process statisticsto investigate properties of the underlying fibre process.

[1] proposed a method to estimate the fibre length distribution from an endpointprocess. We further investigate Ripleys K function and the pair correlation functionto receive information of the fibre length distribution.

References

[1] M. Kuhlmann, C. Redenbach (2015). Estimation of fibre length distributions fromfibre endpoints. Scandinavian Journal of Statistics. 42, (4), 1010-1022 (2015)

Theoretical modeling of biologically inspired compositematerials

Federica Ongaro

School of Engineering and Materials Science (SEMS),Queen Mary University of London, London, UK

Abstract. Many efforts have been devoted to the prediction of the effective propertiesof regular cellular materials with empty cells [1]. In reality, at the microscale, manybiological tissues display a peculiar cellular structure having the internal volumesfilled with fluids, fibers, or other bulk materials. Similar composite solutions are found,for example, in the natural tubular structures like plant stems or animal quills. In thesesystems, the inner honeycomb or foam-like core behaves like an elastic foundationsupporting the dense outer cylindrical shell and makes it more performant [2]. Thoughsome authors numerically and theoretically analyzed the morphology, compositionand mechanical behavior of filled cellular materials, up to date closed-form expressionsfor the effective elastic moduli and constitutive equations have not been derived.To contribute filling this research gap and to provide some useful tools for practicalapplications, I will initially present a continuum model for two-dimensional cellularmaterials having a hexagonal microstructure and the cells filled by a generic elasticmaterial, modeled as a series of closely spaced, independent linear-elastic springs(Winkler foundation). The analysis is then extended to the case of elongated hexagonalcells to study the mechanics of orthotropic composite cellular materials, inspired by thekeel tissue of the ice plant [3]. Finally, in the last part of my talk, I will investigate theeffects of adding structural hierarchy [4] into a composite cellular material. Specifically,I will focus on composite cellular materials composed by structural elements whichthemselves have a structure. The proposed approach is general enough to be appliedto a very wide range of materials, including long fiber reinforced composites.

References

[1] L. J. Gibson, M. F. Ashby: Cellular Solids. Structure and Properties. CambridgeUniversity Press, 2001

[2] L. J. Gibson: Biomechanics of cellular solids, J. Biomech., 36, (2005).[3] L. Guiducci, P. Fratzl, Y. J. M. Brechet, J. W. Dunlop: Pressurized honeycombs as

soft-actuactors: a theoretical study, J. R. Soc. Interface, 11, (2014).[4] R. Lakes: Materials with structural hierarchy, Nature, 361, (1993).

Mircrostructure characterization of fiber reinforcedpolymers based on parametrized space curves derivedfrom µCT data

Pascal Pinter and Kay André Weidenmann

Institute for Applied Materials,Karlsruhe Institute of Technology (KIT), Karlsruhe, Germany

Abstract. Microstructural data is crucial for modelling of fiber reinforced polymers.There are many image features that can be used to describe the fiber architectureusing micro computed tomography analysis. Since it is difficult to track fibers involumetric images, it is even possible to calculate fiber orientations voxel based [1]without knowing about the connection to adjacent voxels. This method is state of theart and is implemented in many commercial toolkits like e.g. VGStudio Max. In recentyears, micro computed tomography has gained image quality intensively. This enablesto acquire high resolution images which are necessary to track fibers correctly fromone end to the other. But most of the currently available software is only applicable forshort fiber reinforced polymers [2] or there is no possibility to evaluate fiber curvature[3].

In this work, an algorithm is introduced to parametrize a space curve for eachfiber within the scanned sample. This method offers the full information about fiberarchitecture and allows for deriving orientation tensors or curvatures directly from theparametrized curves.

References

[1] P. Pinter: Algorithms for the Determination of Orientation-Tensors from ThreeDimensional µCT Images with Various Microstructures. In: International Conferenceon Composite Materials ICCM20 Denmark, Copenhagen, 2015

[2] D. Salaberger, M. Jerabek, T. Koch, J. Kastner: Consideration of Accuracy ofQuantitative X-Ray CT Analyses for Short-Glass-Fibre-Reinforced Polymers, MSF(Materials Science Forum), 907-913 (2015).

[3] M. Teßmann, S. Mohr, S. Gayetskyy, U. Haßler, R. Hanke, G. Greiner: AutomaticDetermination of Fiber-Length Distribution in Composite Material Using 3D CTData, MSF (EURASIP J Adv Signal Process 2010), (2010).

Characterization of a short fiber-reinforced polymer

Céline Röhrig and Stefan Diebels

Lehrstuhl für Technische Mechanik,Universität des Saarlandes, Saarbrücken, Germany

Abstract. This contribution presents ideas, how composite materials can becharacterized with respect to experimental testing. Due to the huge number ofpossible combinations of matrix and filler, each composite material has its owncharacteristics. In this work a polybutylene terephthalate (PBT) reinforced by shortglass fibers is characterized. An overview of the properties of the investigated materialis obtained from results of cyclic uniaxial tensile tests at constant strain rate. The shortfiber reinforced PBT shows elasto-plasticity and damage. These properties depend ona fiber orientation angle, which is investigated by testing specimens with differentangles compared to the main fiber orientation. For the evaluation the complete straininformation in all directions is obtained by an optical area analysis over the wholesurface using a digital image correlation (DIC) software. The optical system uses fourcameras to realize a 360◦-3D-measurement by a two-sided stereography. A final aimof this work is to realize a verification experiment representing the three-dimensionalforming process as realistically as possible. Therefore a deep-drawing based testingdevice, a Nakajima test for polymeric materials is presented and first results of theexperimental investigations are shown. Thereby an optical deformation measurementsystem is also used for the evaluation of the specimen deformation. Finally a first ideafor a one-dimensional modeling description based on the experimental results of theuniaxial tensile tests is presented for further research.

The authors are grateful to the BMBF (Federal Ministry of Education and Research -Bundesministerium für Bildung und Forschung) for financial support through the grantnumber 05M2013-MuSiKo.

Characterization and parameter identification ofmulti-axial behavior in sheet molding compounds

Malte Schemmann and Thomas Böhlke

Chair for Continuum Mechanics,Institute of Engineering Mechanics, Karlsruhe Institute of Technology (KIT), Karlsruhe, Germany

Abstract. Due to their high lightweight potential, economical mass-production andexcellent formability, discontinuous fiber reinforced composites are increasinglyused for nonstructural components in the automotive sector. The application of thisclass of materials is however hindered by a lack of an understanding for robustmodeling the whole process chain as well as the process related thermo-mechanicalprocess-dependent properties. The material class under consideration is SMC (sheetmolding compound), a thermoset matrix reinforced with glass fibers.

The damage behavior of SMC is characterized in uniaxial and biaxial tensile tests,whereas multiple loading ratios are considered to identify damage initiation andevolution. An optimization of the cruciform specimen geometry was performed toobserve damage in the area multiaxial stresses. The multi-objective optimizationcriterion ensures a damage localization in the area of interest instead of the specimenarms, as well an as homogeneous as possible stress state in that area [1].

In experiments the full strain field is measured with digital image correlation. Due tothe inhomogeneity of the stress and strain fields an inverse parameter identificationis required to obtain the material properties [2, 3]. Hereby a Gauss-Newton typeAlgorithm is used to identify parameters of a FEM simulation in a way, such that thesimulated and measured strain fields deviation is minimal.

References

[1] A. Makris, T. Vandenbergh, C. Ramault, D. Van Hemelrijck, E. Lamkanfi, M. VanPaepegem: Shape optimisation of a biaxially loaded cruciform specimen, PolymerTesting, 2010, 216-223

[2] M. Schemmann, B. Brylka, V. Müller, L. Kehrer, T. Böhlke: Mean Field Homoge-nization of Discontinous Fiber Reinforced Polymers and Parameter Identification ofBiaxial Tensile Tests, throught Inverse Modeling, Proc. - 20th Int. Conf. Comp. Mat.,2015.

Comparison of the micromechanical crack orientation inlongfiber-reinforced composites to Puck’s theory

Mario Schmerbauch, Felix Erler, and Anton Matzenmiller

Institute of Mechanics,University of Kassel, Kassel, Germany

Abstract. The micromechanical crack orientation in longfiber-reinforcedcomposites is compared to the crack angle prediction of PUCK’s fracture criterion.PUCK’s model is one of the best performing macroscopic criteria in the World-WideFailure Exercise I [1] and II [2]. As a unique feature among phenomenologicalfailure conditions, it provides the orientation of the fracture plane in the inter-fiberfailure mode for a single ply, which is characterized by a fracture angle. Beforea macroscopic crack occurs in a lamina, cracks evolve on the fiber-matrix level.Hence, a microscopic approach by using a single-fiber Repeating Unit Cell (RUC)is used to investigate this mechanism on the finer scale and its influence on theeffective properties. In this approach, interfaces with an elasto-damage constitutivemodel are inserted into the discretized RUC along the interelement boundaries torepresent matrix cracking and fiber-matrix debonding. The distribution of failedinterface elements is analyzed and compared to PUCK’s results for uniaxial stressstates as well as for in-plane biaxial loading. The microscopic boundary valueproblem is solved with the High-Fidelity Generalized Method of Cells [4].

References

[1] A. S. Kaddour, M. J. Hinton, P. D. Soden: Predictive capabilities of nineteenfailure theories and design methodologies for polymer composites laminates. PartB: Comparison with experiments. In: Failure Criteria in Fibre Reinforced PolymerComposites: The World-Wide Failure Exercise, eds. M. J. Hinton, A. S. Kaddour, P. D.Soden, pp. 1073–1221, Elsevier, Oxford, 2004.

[2] A. S. Kaddour, M. J. Hinton: Maturity of 3D failure criteria for fibre-reinforcedcomposites: Comparison between theories and experiments: Part B of WWFE-II,J. Compos. Mater., 47(6–7), 925–966 (2013).

[3] A. Puck: Festigkeitsanalyse von Faser-Matrix-Laminaten. Carl Hanser Verlag,München, Wien, 1997.

[4] J. Aboudi, M.-J. Pindera, S. M. Arnold: Linear thermoelastic higher-order theory forperiodic multiphase materials, J. Appl. Mech., 68(5), 697–707 (2001).

Generating fiber-filled volume elements with highvolume fraction and prescribed fourth order fiberorientation tensor

Matti Schneider

Department of Flow and Materials SimulationFraunhofer Institute for Industrial Mathematics ITWM, Kaiserslautern, Germany

Abstract. The digital material cycle for fiber reinforced plastics crucially depends onmicrostructures capturing the fiber volume fraction and the fiber orientation. Shortfiber composites used in the industry generally feature a large volume fraction andhigh fiber aspect ratio.In this talk I will introduce a fast and robust method to generate volume elementsincluding straight cylindrical fibers of equal length. It is based on a reformulation ofthe microstructure generation problem for prescribed fourth order fiber orientationtensor in terms of an energy minimization problem, which is then solved numerically.In contrast to existing methods, high accuracy (five significant digits for the fiberorientation tensor), large fiber aspect ratios (up to 150) and large volume fractions(50 volume−% for isotropic orientation and aspect ratio of 30) can be reached. Thetalk closes with a small study on the effective linear elastic properties of the resultingmicrostructures, depending on fiber orientation, volume fraction as well as aspectratio.

Conditions for phenomenological anisotropic damagemodels derived from micromechanics

Stephan Wulfinghoff, Marek Fassin, and Stefanie Reese

Institute of Applied Mechanics,RWTH Aachen, Aachen, Germany

Abstract. The talk investigates the modeling of irreversible anisotropic brittle andductile damage based on second and fourth order damage tensors and its connectionto micromechanical material models. In particular, the conditions for strictly increasingdamage are discussed. Restrictions on the form of the free energy function andthe damage evolution law are proposed. These are motivated by energetical,scale-bridging considerations for growing cracks and pores. Moreover, conclusions aredrawn related to the damage-induced evolution of the yield surface in plasticity.

Error-controlled homogenization for a class of linearelastic disordered materials

Xiaozhe Ju and Rolf Mahnken

Chair of Engineering Mechanics,University of Paderborn, Paderborn, Germany

Abstract. The notion of model adaptivity has been well established, aiming atadaptive selection of mathematical models from a well defined class of models (modelhierarchy) to achieve a preset level of accuracy (see e.g. [1,2]). The present contributionaddresses its application to a class of linear elastic composite problems. We will showthat the classical bounding theories according to [3,4] can provide a model hierarchyin a natural and theoretically consistent manner, without combination of differentmethods using a priori knowledge. As a further benefit, the resulting computationalscheme reduces to a single-scale one. To arrive at computable higher order bounds,the classical singular approximation following [5] is made. As a new finding, thismay, under certain circumstances, give rise to an overlap effect. To overcome this,a correction is proposed. Additionally, the model adaptivity is coupled to the wellestablished adaptive finite element method (FEM), such that both macro model andmacro discretization errors are controlled. The proposed adaptive procedure is drivenby a goal-oriented a posteriori error estimator based on duality techniques. For efficientcomputation of the dual solution, a patch-based recovery technique is proposed, wherea comparison with other existing methods is also given. For illustration, numericalexamples are presented.

References

[1] J. T. Oden, K. S. Vemaganti: Estimation of local modeling error and goal-orientedadaptive modeling of heterogeneous materials: I. Error estimates and adaptivealgorithms, Journal of Computational Physics, 164(1), 22–47 (2000).

[2] F. Larsson, K. Runesson: Adaptive computational meso-macro-scale modeling ofelastic composites, Computer methods in applied mechanics and engineering, 195(4), 324-338 (2006).

[3] P. H. Dederichs, R. Zeller: Variational treatment of the elastic constants ofdisordered materials, Zeitschrift für Physik, 259(2), 103-116 (1973).

[4] E. Kröner: Bounds for effective elastic moduli of disordered materials. Journal of theMechanics and Physics of Solids, 25(2), 137-155 (1977).

[5] A. G. Fokin: Solution of statistical problems in elasticity theory in the singularapproximation, Journal of Applied Mechanics and Technical Physics, 13(1), 85-89 (1972).

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